Statistics (Max Min Problems)

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Try this problem: “A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set? “(A) 32 “(B) 40 “(C) 43 “(D) 50”   Fifteen integers,that’s a little annoying because you don’t literally want to draw 15 blanks for 15 numbers. How can you shortcut this while still making sure that I’m not missing anything or causing myself to make a careless mistake? You could just work backwards: start from the answers and see what works. In this case, you’d want to start with answer (D), 50, since the problem asks for the greatest possible integer. You can also try algebra to solve this problem. On the real test, you generally only have time to try one solution method, so try these both out now to see what you think would work best for you on the test. Check out both solution methods below.

Working Backwards

Ordinarily, you’d start with answer (B) or (D) when working backwards from the answer choices. In this case, though, the problem asks for the greatest possible value, so start with the largest answer choice.
median smallest number (- 25) smallest to largest
(D) 50 25 50 – 25 = 25 25 to 50
The problem specified that the 15 numbers are all different. If that’s the case, then 25 can’t be both the smallest number and the median, or middle, number in the set. Eliminate answer (E) and try (D) next.
median smallest number (- 25) smallest to largest
(D) 50 25 50 – 25 = 25 25 to 50
(C) 43 25 43 – 25 = 18 18 to 43
Can you make 25 the median? List it out. If there are 15 numbers, then 25 should be right in the middle, at position #8. There need to be 7 numbers smaller than 25. 18, 19, 20, 21, 22, 23, 24 There are 7 different numbers all less than 25, so #8 can be 25. There are then another 7 numbers on the other side, the last of which has to be 43 (since the largest number has to be 25 more than the smallest number, 18). The correct answer is (C).

Do the Algebra

Here’s how to answer the question the way we did last week, by “logic-ing” it out via algebra. First, you need to draw out what’s going on, but in some nicer way than drawing out 15 little lines. Here’s what I came up with: gmat2 Next, the range is 25, so the difference between the largest and the smallest is 25. Set the largest to be x (since that’s what they asked for) and the smallest to x – 25: gmat3 In order to maximize x, what do you need to do to the other numbers? In order to maximize x, you need to maximize x – 25 while still allowing it to be the first integer in a series of different integers with a median of 25. In other words, count down from 25, in position #8, to the largest number that you could put in position #1: gmat4 If you feel comfortable counting this out on your fingers, feel free. I think you’d be at least somewhat likely to make a careless error doing that, so you should probably write out the bottom half (the 8 down to 1) and the first number (25). From there, you can  just count it in my head while pointing to each blank. Okay, so the first one is 18 = x – 25, so x = 18 + 25 = 43. Done! The correct answer is (C).

Key Takeaways for Max/Min Problems:

(1) Figure out what variables are “in play”: what can you manipulate in the problem? Some of those variables will need to be maximized and some minimized in order to get to the desired answer. Figure out which is which at each step along the way. (2) Don’t forget to consider other strategies, such as working backwards, when appropriate. On this one, once can argue that working backwards may be easier than going through the max/min steps (at least, it was for me), because the problem dealt with integers and the answer choices weren’t horrible numbers. It was a little lucky that we only had to try two answers, but it wouldn’t have taken that much longer to try the others. (3) Did you make a mistake—maximize when you should have minimized or vice versa? Go through the logic again, step by step, to figure out where you were led astray and why you should have done the opposite of what you did. (This is a good process in general whenever you make a mistake: figure out why you made the mistake you made, as well as how to do the work correctly next time.

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